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Discrete Mathematics 79 (1989/90) 123-146 North-Holland

123

INCIDENCE ALGEBRAS AND COALGEBRAS OF DECOMPOSITION STRUCTURES

Luigi AZZENA Via De Carolis 15, 071(W)Sarsari, Italy

Francesco PIRAS Dipartimento di Matematica dell’ Universita ’ di Cagliari, Via Ospeahle 72, 09100 Cagliari, Italy

Received 23 September 1987

In this paper a few relationships between a Decomposition Structure and its Incidence Coalgebra and Algebra are studied. In particular, some results about the so-called Isomorph- ism Problem for Incidence Algebras of Moebius Categories are generalized. Moreover, we determine conditions under which all endomorphisms and derivations of an Incidence Algebra are continuous with respect to the finite topology.

Introduction

The notion of Incidence Algebra was introduced by G.-C. Rota and others in order to supply a unified algebraic setting for a wide class of problems of enumerative combinatorics. In particular this has made possible a more simple

and general formulation of the classical Mobius Inversion. The structure of Incidence Algebra was originally associated with locally finite posets and monoids with the finite factorization property. Afterwards, this procedure was extended to the set of morphisms of a decomposition-finite category. This includes not only the posets and monoids, but also other objects closely connected to enumerative combinactorics. For the purpose of clarifying the connection between the set S of morphisms of one category and its Incidence Algebra A(S), the study of the so-called Incidence Coalgebra C(S), of which A(S) is the dual algebra, has proved useful. This same notion of coalgebra has furthermore indicated the possibility of utilising this technique in the study of a wider class of objects than that of the categories. One may, in fact, resort to the notion of Incidence Coalgebra whenever one finds a “decomposition structure” S; that is to say, whenever one finds a set S together with a rule, for cutting each of its elements in two, which satisfies certain “reasonable” conditions according to the definition given by Joyal in [7]. In the present work we mean to study the reciprocal relations between S, C(S) and A(S). We observe that in this context the Incidence Coalgebra C(s), in virtue of its relationship of duality with the Incidence Algebra A(S), has an equivalent function to that carried out by the so-called finite topology in A(S).

0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

124 L. Azzena, F. Pim

A classical problem is as follows: to what extent does our knowledge of A(S) or C(S) enable us to reconstruct the decomposition structure S? A few partial solutions to this problem, obtained by Leroux [8] in the case of the categories, are here extended to a class of decomposition structures which we have called regular. In the same way, the Propositions 3.18 and 3.34 are generalizations of analogous results relating to categories and posets (see [8] and [l]). In actual fact such propositions state that all automorphisms and derivations of an Incidence Algebra of a finitely generated decomposition structure are continuous with respect to finite topology. Moreover, we shall see how a simple condition regarding the cardinality of a particular subset S, of S allows us to extend the first of the former propositions to all the endomorphisms of A(S).

1. Decomposition law on a set

Let S be a set. Let N[S X S] denote the free abelian monoid generated by S x S. A decomposition law on S is a pair of applications

d:S-N[SxS]

e:S-N

where the coefficients [&I, usually called section coefficients, satisfy the following equations:

7 iqxt,rJ =? ~,:tlL,svl

T [ Je(d = C [ rsq]e(4) = & , 9 ’

(1.1)

0.2)

The section coefficient [&I counts the different ways in which d cuts the element s ES into the same ordered pair (4, r) ES x S. In the following, the triple S = (S, d, e) will be referred to as a decomposition structure.

Let us denote [q,&, ] the common value of both sides of 1.1. More generally, let us put:

and, for every rz > 1,

L. .“,+,I:=?. [, . . .:,-,, tl[,, in+,1

Incidence algebras and coalgebras 125

Reiterating 1.1, for any sequence of integers 16 ii =Z - - - s ik c n, we have:

If [T,.S.m] > 0, the n-tuple (ri - - - I,) is called a decomposition of degree n of s; [,&] is the number of ways we can cut s into the ordered n-tuple (Ye, . . . , y,).

The decomposition (Ye, . . . , Y,J is called a proper decomposition if d(ri) # (ri, yi)

for 1 s i sn. The supremum, in N U {m}, of the set of degrees of proper decompositions of an element s E S is called the length of s and denoted I(s). It is easy to check that I(s) = 0 if and only if d(s) = (s, s). For every n E N we put:

S,,,:={s~S]1(s)=n} and Sn:=kQnSCnj.

Owing to the following proposition S,, together with the restriction of the decomposition law, may be considered as a “substructure” S,, of S.

Proposition 1.3. Zf [&I > 0, then l(q) + I(Y) G f(s).

Proof. Observe that if (ql, . . . , qh) and (Ye, . . . , yk) are proper decompositions of q and r, respectively, then

41 . .

As a consequence (ql . - . qh, Y, . . * yk) is a proper decomposition of s. Cl

In the study of the decomposition structure S, a central role is played by the neutral elements (i.e. the elements s E S such that e(s) = l), as shown by the following propositions, due essentially to Joyal [7].

Proposition 1.4. For each s E S, there exists a unique pair of neutral elements a,(s) and a,(s) such that both [ &J.J and [o~c~J aye positive.

We have:

Lo(Z), sl= L,;(s)1 = l. Proposition 1.5. Zf s E S,, then s is a neutral element of S.

126 L. Azzena, F. Piras

Proposition 1.6. The following statements are equivalent: (i) s is neutral;

(ii) e(s) > 0; (iii) a,(s) = s (resp. a,(s) = s).

Proposition 1.7. Zf [&I >O and e(r) = 1 (resp. e(q) = l), then s = q and r = a,(s) (resp. s = r and q = E&(s)).

Proposition 1.8. Zf[&]>O, then h(s)= h(q), al(q)= h(r), 4(r)= 316).

If q, r are neutral elements and U is a subset of S then we put:

U(q, r) = {s E U 1 a,(s) = q and a,(s) = r}.

The Zength n graph of a decomposition structure S is the direct graph whose vertices are the neutral elements of S and whose edge-set is SCn,. The arrow s E S(n) is directed from G’,(s) to a,(s). Assuming the whole S as edge-set, we get a new directed graph, the so-called associated graph of S. Obviously, the associated graphs of two decomposition structures S and B are isomorphic if and only if there exists a bijection 8 : S+ T such that 6(&(s)) = &(0(s)) and

e(&(s)) = 4(%)), f or every s ES. Moreover if r(e(s)) = Z(s) for every s ES, then we say that S and U have isomorphic presentations. In this case S and T have isomorphic length n graphs for every n E N.

A decomposition structure S is said to be hereditarily finite if each s E S admits a finite number of proper decompositions.

Proposition 1.9. The following statements are equivalent: (i) every element of !S has a finite length;

(ii) S is a hereditarily finite decomposition structure; (iii) every neutral element has length zero; moreover if, for some s, [rsS] > 0 then

r is a neutral element.

Proof. (i) j (ii). It is sufficient to show that, for each s E S, the number decompositions of any degree is finite. Obviously, this is true for degree Arguing by induction, from

of 2.

we obtain that the number of sequences (rI, . . . , r-,+1) such that [,,.._sY+,] > 0 is finite.

(ii)+ (iii). (See also [7], Theorem 7, p. 67). Let s be a neutral element; if

Incidence algebras and coalgebras

d(s) f (s, s), then from

127

we deduce that (s, . . . , s) is a proper decomposition of s. Thus s admits proper

decompositions of any length. This contradicts (ii). Suppose, now, [rss] > 0. If s is a neutral element then d(s) = (s, s); so r = s. On the other hand, if neither s nor r are neutral elements then

It follows that s admits proper decompositions of any length which again contradicts (ii).

(iii) 3 (i). Let m denote the number of decompositions of degree 2 of an

element s E S and let (rI, . . . , r,J be a proper decomposition of s. Since

there exists q2 E S such that

q2 [ 1 >O and

s 1

> 0. rl, r2 q2, r3 - * . r,

Repeating the same argument one obtains a sequence ql, . . . , qn-l of elements of S such that ql=rl, [4:;+,]>0, lsisn-2, and [,,,,,” ,... J>O, l~isn-1. Similarly one may obtain a sequence pl, . . . , P,,-~ such that [/,,I > 0 and [,;+fri.,] > 0, 1 G i c n - 1. Since i >j implies [4,,r,y;.._r,] > 0, all this proves that there exists an element tji such that [$J > 0 as well as [,+K.,] ~0. Since d(rJ # (rr, t-J, from (iii) we deduce d(tji) # (tji, tji) and qi # qj* Therefore the pairs (qi, pi) get n - 1 distinct decompositions of s. Thus rz G m + 1. Cl

A decomposition structure is associated with any small category C such that the set {(q, r) 1 q, r E Mar(C), qor = s} is finite for every s E Mar(C). The decomposition law is now:

d:S--N[SxS]

S* C br) ‘“4 =s

e:S-N

ifs is an identity of C

otherwise

128 L. Azzena. F. Piras

When such a decomposition structure is hereditarily finite, C is said to be a Miibius Category. Decomposition structures associated with small categories have been studied in [3, 81.

We shall see that many results obtained for Mobius Categories are also true for a larger class of decomposition structures, the class of regular decomposition structures.

Let s = (S, d, e) be a decomposition structure and Z[A] denote the free E-module generated by the set A. Let us put

where the sum ranges over all proper decompositions (ri, . . . , r,,+J of degree n + 1 of s.

S is said to be an n-regular decomposition structure if Ker(@) = Z[S,]. Moreover, s is said to be a regular decomposition structure if s is an n-regular decomposition structure for each n E N.

All the decomposition structures associated with small categories are regular. S is said to be a finitely generated decomposition structure if S is a regular

hereditarily finite decomposition structure and, for each pair q, r E S,,,

(i) S&q, r) is finite; (ii) [q, r] = {s E SO ( th ere exist U, v, w E S with [“:,,,I > 0 and a,(u) = q, G’,(w) = s

and a,(u) = r} is finite.

Proposition 1.10. If S is a finitely generated decomposition structure then, for each

pair q, r E S,,, the set S(,,(q, r) is finite.

Proof. The proposition is trivial for n = 1. Let us prove it for n + 1 under the assumption that it is true for n. To this aim, consider the set U of all proper decompositions of degree n + 1 of the elements of S,,+,,(q, r). Notice that, because of the regularity of S, the cardinality of S,,+,,(q, r) is less than that of U. Let (ri, . . . , r,+I) be a proper decomposition of s E Sc,+lj(q, r). Since

there exist u E [q, r] and u E S(,,(q, u) such that r,+l E ScIj(u, r) and [r,.Y.r,] > 0. By inductive hypothesis, we deduce that for every u E [q, r] the set of the proper decompositions of all the elements of S,,,(q, u) is finite. It follows that U, and then S,,+,,(q, r), is finite. 0


2. Incidence coalgebras

Let K be a field of characteristic zero. Each decomposition structure S = (S, d, e) allows us to define a coalgebra over K. Let us associate a variable x, to each s E S and denote K[S] the K-vector space spanned by x,‘s. Owing to 1.1 and 1.2 the linear maps:

As : K[S] - WI @ K[Sl

&s:K[S]- K

x,-e(s)

(2.1)

(2.2)

can be considered as a diagonalization and counit map respectively in a coalgebra structure (see [lo]). This coalgebra, denoted C(s) = (K[S], A,, Ed), is said to be the Incidence Coalgebra of S.

By Proposition 1.3, we have A,(K[S,]) E K[S,] @ K[S,J for each n E N. Thus if we restrict both As and .sS to K[&], we obtain a subcoalgebra C(S,) of C(s). Another class of subcoalgebras of C(S) is obtained considering, for each pair p, q E SO, subsets of S of the kind S(,+,,(p, q) US,.

Consider the maps:

n+l

A; : K[S] - G3 MS1

and n+l

2”s: K[S] - @ K[Sl

where the last sum ranges over all the proper decompositions of degree n + 1 of S. Obviously, A, = Ai and 6: is the restriction of dg to Z[S]. We also have:

where Ps denotes the linear map

Ps : K[S] - WI 0 ifseS,

XsH 1 otherwise


It is plain that if s is an n-regular decomposition structure then

Ker(&) = K[&]. An element c E C(S) is said to be a grouplike element if As(c) = c 63 c and

Q(C) = 1. The following proposition tells us that neutral elements of a hereditarily finite decomposition structure S are precisely the grouplike elements of the associated coalgebra C(s).

Proposition 2.3. Let s be a hereditarily finite decomposition structure. An element

c E C(s) is grouplike if and only if c = x, for s E S,.

Proof, Ifs E S, then X, is a grouplike element of C(s). Conversely, let c = C, k”x, be a grouplike element of C(S). We have:

xq 60 x, = c k”As(x,) = As(c) = c 0 c = c k9krx, @IX,. s 49’

Thus C, k”[&] = k9k’ for each pair q, r E S. Since s is a hereditarily finite decomposition structure, max{l(s) 1 x, occurs in c} is a non-negative integer n.

So, if x, occurs in c and l(t) = n then C, k”[&] = k’k’ f 0. Hence, there exists u E S such that k” f 0 and [t] # 0. It follows by Proposition 1.3, I(u) 2 2n, i.e. n = 0;

so if x, occurs in c then u E S,. Since grouplike elements of a coalgebra over a field are linearly independent (see [lo]), we have c = k”x,, s E S,,. Owing to EJC) = 1, we deduce k” = 1. This completes the proof. 0

Let C(s), C(U) be Incidence Coalgebras. We recall that a linear map

$J : C(S) * C(U) is a coalgebra map if A, 0 @ = ($8 $) 0 As and .sT 0 Q, = .sS. From this definition, by induction on n, we deduce:

A similar formula holds for d” only if T is a hereditarily finite decomposition structure. We shall prove this after the following proposition.

Proposition 2.4. Let T be a hereditarily finite decomposition structure. Zf

qI : C(S) + C(T) IS a coalgebra map then, for every s E S,, there existi t E To such

that $(xs) = x,.

Proof. Let s E &; then &(9(xs)) = (@ @ #) o A&) = cP(%) @ +(-G) and +(4G)) = s&J = 1. Thus $(xs) . IS a grouplike element of C(U). Hence, by Proposition 2.3, #(x,~) =x, where t E T,. 0

As a consequence of Proposition 2.4, if U is a hereditarily finite decomposition structure, every coalgebra map # : C(S)-, C(U) gives rise to a map &: S,+ To

Incidence algebra and coalgebras 131

where &,(s) = t e $~(a) =x,. In the following we shall always denote & the associated map with the coalgebra map $C

Corolkuy 2.5. Let U be a hereditarily finite decomposition structure. If

c#J : C(S)-+ C(U) is a coalgebra map then

Proof.

Now, by Proposition 2.4 and by the definition of PT, (Bn+’ (PTo@))o(Ag - &)

is the zero map. Thus A;,# = (Bn+’ PT)o(Bn+’ #)06:. 0

Proposition 2.6. Let U &e a hereditarily finite decomposition structure and

@ : C(S)+- C(T) a coalgebra map. Zf Z(s) is finite and if xI, t E T, occurs in @(_xs)

then there existi a decomposition (p, q, r) of s such that x, occurs in #(x,) and

~cl(t) = 4kl(M4))~ 4(t) = &l(4(q)).

Proof. Notice that if we put @(xS) = CIET &xx, for each s E S, from

($8 4 @ 49(&J) = A%@(x,))

we obtain:

for each s E S and each triple LX, /I, y E T. So, assuming (Y = a,(t), p = t, and y = a,(t), we have:

Now by Proposition 2.4, the proposition holds when I(s) = 0. Arguing by induction on I(s) = n, let us assume that it holds for the elements of S with a length less than n. Given the former equality, if &#O there exists a decomposition (u, V, w) of s such that @$‘(‘) # 0, @: # 0 and #$“‘f 0. If u = a,(s) and w = a,(s) then u =s; thus, by Proposition 2.4 6’,(t) = &,(&(s)) and e,(t) =

@,,(E+(s)). Otherwise I(V) <I(s), and, by induction hypothesis. there exists a decomposition (i, q, j) of v such that x, occurs in @(x,), a,,(t) = &(2+,(q)) and

W) = 90(W)). w e obtain the proof of the existence of the decomposition


(p, q, r) of s by:

Proposition 2.7. Let # : C(S)-, C(U) b e a coalgebra map. Zf U is an n-regular hereditarily finite decomposition structure then, for each m < n, @(C($&)) G

C(U,).

Proof. The special case n = 0 has been considered in Proposition 2.4. Now let us

suppose n > 0. If X, E K[S,,,], m G n, then dg(x,) = 0. Hence, by Corollary 2.5,

&(@(x,)) = 0. Th us, owing to the n-regularity of U, #(xs) E K[T,,]. 0

Corollary 2.8. Under the hypotheses of Proposition 2.7, if x, occurs in #(xs) and Z(t) 3 Z(s) = n + 1 then a,,(t) = &(&(s)) and 6’,(t) = &(dI(s)).

Proof. Since l(s) is finite and x, occurs in +(xs), by Proposition 2.6, there exists a

decomposition (p, q, r) of s such that x, occurs in $(x,). Thus, by Proposition

2.7, I(q) 2 n + 1. Therefore q = s and, by Proposition 2.6, we get the thesis. 0

Combining Proposition 2.7 and Corollary 2.8 one can easily prove the following

result.

Proposition 2.9. Let $: C(S)-, C(U) b e a coalgebra map. Zf U is both an n-regular and (n + 1)-regular hereditarily finite decomposition structure then, for each pair p, q ES,,,

f$(C(Sc,+,,(p, q) u S)) G C(~@l+I,(MP)~ @o(q)) ” TJ.

Proof. Since U is both an n-regular and (n + 1)-regular decomposition structure,

by Proposition 2.7, $(C(S,+,)) E C(U,+,) and #(C(%)) c C(U,). Moreover, by

Corollary 2.8, ifs E S,,+,,(p, q) then

$(xs) e C(~,,+I,(~o(P)~ #o(q)) u T?l)

and the thesis holds. 0

Corollary 2.10. Let @ : C(S) +- C(U) b e a coalgebra isomorphism. Zf 55, U are both n-regular and (n + 1)-regular hereditarily finite decomposition structures then, for every pair p, q E S o, the cardinal& of S,,+,,(p, q) is equal to the cardinal@ of

T@l+I,(@o(P)9 @o(q))*

Proof. By Proposition 2.9, C(S,,+l,(p, q) U W = ~(U,,,+I,(@O(P), b(q)) U a,).

By Proposition 2.7, C&J = C(U,). Hence the K-spaces

ZW,,+,,(p, 411 = K&,+I,(P, q) u WKKLI


have the same dimension. Cl

Now we are able to state the main result of this section.

Proposition 2.11. Let S, % be a regular hereditarily finite decomposition structures. Zf C(S) and C(U) are isomorphic coalgebras then S and T have isomorphic presentations.

We observe that the hypotheses of n-regularity and (n + 1)-regularity are necessary in Corollary 2.10 as the following example shows.

Let S = (~4, u2, 243, p, q, r, s>. If SO = (4, u2, h), S(I) = {p, 4, r>, s(2) = {s)

with u1 = d,(p) = a,(r) = a,(s), u2 = d,(p) = do(q), 2.43 = 4(q) = a,(r) = 4(s) and [pSq] =2 then S is a regular decomposition structure. While, if T =

(4, 4 4, p’, q’, I’, s’), To = (4, u;, 4, qlj = {p’, q’>, q2) = {I’, 0, 4 =

d,(p’) = a,(r’) = a,(s’), u; = d,(p’) = d,(q’), u; = aI = aI = a,(~‘) and [,&I = [,[;,I = 1 th en U is not a l-regular decomposition structure. For

x,* -x,, E Ker(6;) and x,, - _r,, $ Z[ T,]. Obviously .S and U do not have isomor-

phic presentations while the linear map @ : C(S)+ C(U) defined by ~#J(x~,) = n,:, i = 1, 2, 3, &x,) =x,., #(x,) =xq., @(xS) =xSP +x,, and $(xr) =xSz -x,, is a coalgebra isomorphism.

3. Incidence algebras

Let R be a commutative ring and A, B two R-modules. A family (J)iel of elements of Hom,(A, B) is said to be a summable family if, for every a E A, the set {i E I 1 f;(a) # 0) is finite. Given a summable family (fi)isr of elements of Horn&A, B) we obtain a new element Cie,A of Hom,(A, B) putting, for each aeA,

( > 2.J (a):= 25(a).

Thus, if A = R[S] and B = R the set (x~)$~~, where

Y:R[S]- R

1

1 ifr=s XrH 0 otherwise

is a summable family. Obviously, if f is an arbitrary element of R[S]* =

Hom,(R[S], R) then the family (f(xS)xS),,, is a summable family and we have:


Proposition 3.1. Let ‘1’ : R[ T]* + R[S]* be a linear map. Then the following statements are equivalent:

(i) there exists a linear map #:R[S]+ R[T] such that q(f) =f OC#I (i.e. q tk the dual map of @);

(ii) if the family (f;:)isr is a summable family, then the family (alar is also a summable famiZy and q(Ci,rf) = Ciet q(L);

(iii) the family (V(X’))H is a summable family and, for each f E R[ T] *,

l/J(f) = CM-f (x&(x?

If we consider R as a topological ring with the discrete topology then R[S]* is

naturally provided with a structure of topological module by the product topology

(or finite topology). According to this scheme of things Proposition 3.1 can be

restated in the following way:

Proposition 3.2. A linear map 1/, : R [ T] *+- R[S]* is continuous, with respect to finite topology, if and only if 111 is the dual map of a linear map C$ : R[S]-+ R[T].

We shall assume throughout that R is a characteristic zero field K. We recall

that the Incidence Algebra of a decomposition structure s is the dual algebra of

the Incidence Coalgebra C(s). Thus, the product f *g of the elements f, g E K[S]* is given by

(f *g)(xJ:=mO(f @gPAs(xJ = z [qsrlf(xJg(x29

where m : K @I K+ K is the product over K. It is plain that this product is associative and the linear map cs is the two-sided

identity. The Incidence Algebra of S will be denoted A(s). We begin our study of the algebra A(s) by considering two families of its

ideals.

Let X be a subset of S, denoted by d(X) the set of the elements of S which are

in at least one decomposition of an element of X, then

d(X)I:={f EA(F~) If(x,)=O forevery sod}

is a two-sided ideal of A(!%). Another family of two-sided ideals of A(s), which we need to consider is, for

each n E N,

J@) := {f EA(% 1 s E s,-13 f (xs) = O>

Moreover we have:

A(s) = J,,(S) z J,(s) 2 . * - and (Ji(s))” c J,(s).

It is easy to check that the quotient algebra A(s)/.Z,(F5) is isomorphic to the


algebra A(S,) obtained providing K[&]* with the product

We observe that the family (x’),,~ is the unique maximal family of non-zero orthogonal primitive idempotent elements of A(s,).

The existence of the ideals d(X)l and J,,(S) as well as of the algebra A($$,) allows us to state that if we consider K provided with the discrete topology and A(S) with the product topology then A(S) is an Abstract Incidence Algebra over K according to the definition given by Di_ir in [5]. In this case H(O) =A@,), H,, =J,($G) and we obtain a O-basis of two-sided ideals considering the family d(X)l when X is a finite subset of S.

If s is a hereditarily finite decomposition structure and f E Ji(s) then the family

(f %eN is a summable family. As a consequence it is possible to prove (see [7]) the following statement about the invertible elements of A(s).

Proposition 3.3. Zf S is a hereditarily finite decomposition structure then an element f E A(s) is invertible if and only if f ( ,) x is an invertible element of K for each s E So.

Corollary 3.4. J,(S) is the Jacobson radical of A(s).

For our purposes the knowledge of the inter-relation between the ideals Jn(s) and J1(s))” is fundamental. We already know that (J1(s))” E Jn(s) but, generally, Jn(S) & (J1(S))n. That is especially evident when s is not an n-regular decomposition structure. In fact, the following proposition shows that the set {f E

Jn+dS) If&) =O f or all but a finite number of s ES} is not contained in

(JIW) H’ if s is not an n-regular decomposition structure.

Proposition 3.5. The following statements are equivalent: (i) S is an n-regular decomposition structure;

k (ii) for every pair of finite sequences s,, . . . , s, E S, f(si) > n, and . . . ) k, E K there exists a finite family of scalars h, ,... r”+,, with (rI, . . . , r,+I)

pioper decomposition of degree n + 1 of si, 1 G i s m, such that h(x,) = ki, where h = C h,,...,+,x” * . . . *.P+‘;

(iii) for every pair of finite sequences s,, . . . , s, E S, l(s,) > n, and k I,..., k, E K there exists f E (J,(S))n+’ such that f (xs,) = ki, 1s i 6 m.

Proof. (i) + (ii). If S is an n-regular decomposition structure, then the K-space spanned by d”,(x,)‘s, I@,) > n and 1 s i cm, does not have dimension less than m. Hence, if &(x,) = C [T,..S;n+,]xT, @ * * * @x,,+,, there exist h ,,... r,+, such that

136

C [,..:;_+,]h, ,...,” +1 = k. Thus

L. Azzena, F. Piras

h = 2 h,,...,“+,xr’ * - - . * xrn+’

satisfies the conditions h(x,) = ki.

(ii) j (iii). This is trivial. (iii) + (i). If s is not an n-regular decomposition structure there exist a finite

sequence sl, . . . , s, of elements of S, I@,) > n, and a finite sequence h,, . . . , h, of elements of K such that czl hidg(x,,) = 0. Thus Cz”=, h,[,,..?;,+l] = 0 for every

r1, . . .,r,,+leS-So. Now,ifg=g,*.-. *g,+l, gj E./~(S), 1 ~j G n + 1, then

2 hig(xs,) = 2 hi C [ rl i=l i=l

SirH+l]gl(Xr,) . * * gn+l(Xr,+,) . . .

= C (g hi[ rl . eS:r,+l])gl(xr,) . . ’ &+l(xrn+,) =O.

So, for every f e (J1(s))n+l, we have CEi hif(x,,) = 0. Therefore, if kl, . . . , k, is a sequence of elements of K with CE1 hiki #O there exists no element

f E cww” such that f (xs,) = ki, 1 c i sm. Cl

As a consequence of the former statement we see that if the set {si E S 1 l(Si) > n}

is finite and S is an n-regular decomposition structure then (.Ji(s))” =J,,(s).

However, also under the hypothesis of n-regularity for s, generally, (J1(5G))” # J,(s). The only thing that we can state, by Proposition 3.5, is that if we consider A(S) equipped with the finite topology then s is an n-regular decomposition structure if and only if J,(s) is the topological closure of (.&(s))=. Nevertheless, if S is a finitely generated decomposition structure we can prove a “local” equality

between (Ji(s))” and J,(s).

Proposition 3.6. Zf S is a finitely generated decomposition structure, then, for each

n > 1 and for each pair p, q E S,,

xp *J,(S) *xp = xr * (Jl(S))” *x4.

Proof. It is sufficient to prove that xp *J,(S) *x4 E xp * (.J1(s))” *x9. The proof is by induction on n. The proposition clearly holds when n = 1. Thus, we suppose the conclusion holds for n. By Proposition 1.10, since .!5 is a finitely generated decomposition structure, the sets S(,,(p, q) are finite for each m 2 1. Suppose f EXJ’ *Jn+l(S) *x9. By Proposition 3.5, we can find a sequence of linear forms

Incidence algebras and coalgebrm 137

g,+i E (J1(S))n+i, i = 1, 2, . . . , such that

g,+&) =f&) for every r E %+,j(p, 4)

i-l

g,+i(&) =f(-%> - C &Yrz+jCxr) for every r E s(n+i)(P9 9)

j=l

where g,+i = C h,,...,n+z~” * * . . *Yn+’ and the sum ranges over all the proper decompositions of degree n + i of the elements of S,,+,,(p, q). Obviously, the family (g,+i)iS1 is a summable family and we have f = Xi&i g,+i. Therefore:

f = c ucb.ql

(xv* (c h,...rfl+,xr2*. . - *P+$)

uql)(P+)

where the second sum ranges over all the proper decompositions (r-i, . . . , rn+j) of the elements of Uial S,,+,,(p, q) with rl = V. Thus

f = x xv*gv,, ~+wl

“+)(P.U)

with g,,, EXJ’(U)*Jn(s) *x q. So, by induction hypothesis, and, since the set of pairs (u, V) such that u E [p, q] and u E So,(p, U) is finite, we can conclude that f EXP * &(s))n+l *x4. 0

We now come to the study of the algebra maps between two Incidence Algebras and their relationships with the coalgebra maps. With reference to this we observe that to be sure that an algebra map r/.~ is the dual of a coalgebra map it is sufficient to make sure that r/~ is the dual of a linear map. In fact, the following proposition holds.

Proposition 3.7. If the algebra map I$:A(T)-A(S) is dual of a linear map @ : C(s)+- C(U) then C#I is a coalgebra map.

Consequently, by Proposition 3.1, we have;

Proposition 3.8. Let T/J :A(U)-,A(F$ be an algebra map. If (i) the family (1+9(x’)) tsT is a summable family ;

(ii) for each family (kJrsT of elements of K Vf(L,~kx’) = &~kW(x’); then ‘1’ is dual of a coalgebra map # : C(s) * C(U).

In the following, owing to Proposition 3.2, if the algebra map w satisfies the hypotheses of Proposition 3.8 we shall say that r,9 is a continuous algebra map.

Corollary 3.9. Every inner automorphism of A(s) is a continuous automorphism.


We begin our study of the algebra maps with a simple proposition. This result has been proved by Leroux (see [S]) in the particular case of the Mobius Categories, but it is possible to repeat the same proof in our case.

Proposition 3.10. Let T be a hereditarily finite decomposition structure. Zf

q:A(T)*A(S) is an algebra map then

QWU)) G W).

Proposition 3.11. Let T be a hereditarily finite decomposition structure and

q:A(T)+A(s) an algebra map. Zf I( ) * fi ‘t s IS nt e and if q~(x’)(x~) # 0 then there

exists a decomposition (u, u, w) of s such that ~(x’)(x,,) # 0 and ~(x”~(“)(x~~~,,,) =

ly(xaqx~,~v,) = 1.

Proof. If I(s) = 0 then s = a,(s) = a,(s). From I+v(x’)(x,) # 0 we get:

+(x&Kr) *Xf *Xa,(0 )(xJ = Ill(~“““‘)(~s)~(~‘)(~s)llt(~“l”‘)(~s) + 0.

So, ~(x”~“‘)(x~) = k # 0 and, since

W(X&l(Q )(4 = ~(~ao’r’)(~,)~(~ao~f~)(~,),

we have k = 1. Let us suppose, now, that the proposition holds for every element of S with length less than m. If I(s) = m we have:

V(X’)(%) = a:= [a ; c]~(~J~“‘)(~.)w(~‘)(~~)~(~~~~~~)(x”””)Q) f 0. 9 7

Consequently, there exists a decomposition (a, b, c) of s such that r+!~(x”““‘)(x,) # 0, r+Q(Xf)(Xb) # 0, rjJ(x”l”‘)(Xc) # 0. If a = 6’,(s) and c = G’,(s) then b = s and the proof is concluded. Otherwise l(b) <m and, by induction hypothesis, there exists a decomposition (4, u, r) of b such that

V(x JoW)(~ Jo& = ~(x”‘(‘%xJ,~~~) = 1 and V(x%) # 0

Now, as in Proposition 2.6, we have:

Consequently there exists a decomposition (u, V, w) of s such that

V(X Jo(o)(x +,(,,) = q(~~~(~))(x~,(,J = 1 and vWk) f 0. 0

Corollary 3.12. Let !?5, U be hereditarily finite decomposition structures. Zf

q~ :A(U)+A(s) is an isomorphism then, for each s E S,, there exists t E To such

that I = 1.

Proof. Since 3 is onto, for every s l S,, there exists an element f = CreTfs’ of A(U) such that q(f)= x’. Let X = {t E T 1 fr # 0); by Proposition 3.10 f $ J,(U),


therefore X rl 7’,‘,#0. If t E X II T, then ~‘*f = xf + CveYfuxU where Y=

{V E X 1 Z(V) > 0 and a,(v) = t}. Thus

Since lu(C,,,fvx”) E 4(%, we find that if r E S, and r #s then I/.J(x’)(x~) = 0. But t/~(x’) # 0, therefore there exists w E S such that r/~(x’)(x,,,) # 0. So, by Proposition 3.11, q(x’)(xs) = 1. 0

Proposition 3.13. Let s, T be hereditarily finite decomposition structures and +:A(T)+A(S) an algebra map. Zf, for every s E S,, there exists t E To such that I+!J(x’)(x~) = 1 then there exists an inner automorphism L of A(s) such that, for every q E To (10 I) = 0 whenever l(r) > 0.

Proof. At first we observe that if s E S, and there exists at most one t E To such that IJJ(x’)(x~) = 1. In fact, if u E T,, u #t, and r/~(x”)(x~) = 1 then 1 =

W)(xJW%) = V( x’ * x”)(x,) = 0. Consequently, by hypothesis, for each s E S, there exists a unique t, E To such that r/~(x&)(x~) = 1. Since (q(x”) *x’),,~ is a summable family of A(S) putting g = &, r@(x”) *xs we obtain an invertible element of A(s). If t E T,, then

Therefore

g-i* l#(x’) *g = c xs. (sGllG=t)

Thus the map

&:A@)- A(S)

h-g-‘*h*g

satisfies our requirements. Cl

If we limit ourselves to consider Incidence Algebras of finitely generated decomposition structures it is possible to give more detailed results. Particularly, under this hypothesis, it is possible to find conditions such that an algebra map is continuous and to prove a proposition analogous to Proposition 2.11.

Proposition 3.14. Let T be a finitely generated decomposition structure. Zf v:A(T)+A(s) is an algebra map then, for each u, v E T,,

q(x” *J,(U) *x”) E (J*(S))“.

140 L. Azzena. F. Pirm

Proof. If u, u E To then, by Propositions 3.6 and 3.10, we have:

qJ(x” *Z,(U) *x”) = q(x” * (J,(U))” *Xv) E qJ(XU) * (.J@))” * qqx”)

= (.Zi(q)n. cl

Corollary 3.15. Zf t E T, s E S and I$(x’)(x~) # 0 then l(t) <l(s).

Proof. If Z(t) = n then x1 EX W) *J”(U) *xal@). So, by the proposition, if Z(s) <Z(t)

then I = 0. Cl.

Corollary 3.16. Zf .S is hereditarily finite, then the family ( ~I(x’))~~~ is a summable family of elements of A(s).

Proof. Let T, = {t E T 1 v(x’)(x~) f 0) and let TG = {(p, q) 1 p = &@), q = 4(t) andtET,}. IfseS,, since the number of decompositions (u, u, W) of s is finite,

by Proposition 3.11, we see that Tos is finite. Now, by Corollary 3.15,

T, G UWETc& T,(p, q); there fore, by Proposition 1.10, we can conclude that T,

is finite. 0

Proposition 3.17. Let T be a finitely generated decomposition structure and let s be a hereditarily finite decomposition structure. An algebra map v : A(U) --;, A(S) is continuous if and only if, for each s E S,, there exists t E To such that I = 1.

Proof. If q?:A(U)+A(S) is dual of a coalgebra map #: C(s)+ C(U) then

q(x’) = xf 0 #. Now, by Proposition 2.4, for every s E S, there exists t E To such

that @(xs) =x,. So, for every s E S, there exists t E I&, such that q~(x’)(x~) =

(x’o @)(xs) = xf(x,) = 1.

Vice versa let us suppose that, for each s E S,, there exists t E To such that

I = 1. In this case, by Proposition 3.13, there exists an inner automorph-

ism L of A(s) such that, for each q E T,, (L 0 W)(xq)(xr) = 0 whenever l(r) > 0. Now, by Corollary 3.9, L is continuous; so to conclude the proof, it is sufficient to

prove that L 0 v is continuous. By Corollary 3.16, ((Lo+)(x’)),,~ is a summable

family. Therefore we must only prove that, for each family (kt)r.T of elements of

K and for each w E S, we have:

(1 O +~kx’)(xW) = lT/& o Ilt)(x%v).

Let w E S,. If we put a,(w) = u, G’,(w) = 21 and if we denote t,, t, the elements of

To such that q(x”)(xJ = I = 1 then we have:


thus, since T,(t,, t,,) is finite, we have:

W,+ k-+4 = c ML o luW)(xw) = c 41 o wW&w). 0 teT ~~Trn(~u~~u) IET

As a consequence of the former statement we obtain a result which generalizes Leroux’s proposition about the Isomorphism Problem for Incidence Algebras of Mobius Categories (see [8]).

Corollary 3.18. Let 54 % be finitely generated decomposition structures. Every algebra isomorphism v : A(U) + A(s) is continuous.

Proof. By Corollary 3.12 and Proposition 3.17. 0

Corollary 3.19. Let $$ T be finitely generated decomposition structures. Zf A(s) and A(U) are isomorphic Incidence Algebras then ~3 and U have isomorphic presentations.

Proof. If A(U) -A(s) then, by Corollary 3.18, C(S) = C(U). Thus, by Proposi- tion 2.11, we come to the conclusion. 0

Proposition 3.17 shows that if U is a finitely generated decomposition structure then the occurrence that $J is continuous results from its behaviour on the set TO. This is made more evident by the following proposition.

Proposition 3.20. Let U be a finitely generated decomposition structure and let !5 be a hereditarily finite decomposition structure. An algebra map I$ : A(U)* A(s) is continuous if and only if is continuous the algebra map

w : 4-WW) - 4W4W

f + Z,(U) H V(f) + ZIW

Therefore, now we can restrict our analysis, about the relationships between algebra and coalgebra maps, only to include the algebra maps between A(U,) and A($$,). The main instrument, which we shall use, is the notion of ultrafilter in the lattice P(T,) of all the subsets of TO. This is related to the following proposition.

Proposition 3.21. Let I/J :A(UO)-+A(S ) O an algebra map. Then for each s E SO the set

Us = {X c To 1 w( c +s, = 1) fEX

is an ultrafilter in P(TO).

142 L. Arzena. F. Piras

Therefore the following proposition holds:

Proposition 3.22. An algebra map T+Q : A(T,) + A(S,J is continuous if and only if, for each s E S,, IF, i3 a principal ultrufilter in P(T,).

Corollary 3.18 assures us that every isomorphism between Incidence Algebras of finitely generated decomposition structures is always dual of a coalgebra map. The following propositions establish conditions on To in order that every algebra map ~:A(UO)-+A(S,) is dual of a coalgebra map.

Proposition 3.23. Let I/J :A(%,)* A(&,) be an algebra map. Then

VGT, k&)(q) = 0 if and only if {t E To 1 k, = 0} E FS.

Proof. Let us suppose q(CfeTo k$)(q) = 0. If X = {t E To 1 k, = 0} $ F, then, putting X’ = {t E & ( k, ZO}, we have:

q@, k$)(x&@, k;‘x’)(x,) = W( c xt)(xS) = 1, since X’ E FS. tCX’

Therefore $J(&, k,x’)(x,) = $J(&,,. k$)(q) # 0. Vice versa suppose X E 5,.

Then rj$Zre~ @7(x,) = NZ,,F k~‘)(x,)~(C,,,x’)(x,) =O. 0

We recall (see [2]) that, given an arbitrary infinite cardinal a, an ultrafilter ff is said to be &-complete if and only if the intersection of any set of fewer than a! elements of IF belongs to 5. The cu-complete ultrafilters are characterized by the following lemma.

Lemma 3.24. Let I be u set and let F be an ultrajilter in P(Z). F is a-complete if and only if for every partition of Z into fewer than a parts, one of the parts belongs to ff.

Proof. See [2] p. 180. Cl

Proposition 3.25. Every algebra map ly : A(%,)+ A(!S,) is continuous if and only if there exists no IK(+-complete non-principal ultrafilter in P( T,). (IK(’ denotes the least cardinal greater than the cardinal 1 KI of the jield K).

Proof. Let F be a IKK(+-complete non principal ultrafilter in P(T,). By Lemma 3.24, for each map f : To+ K there exist X, E F and k, E K such that f (t) = k, for each t E X,. Now, let s be a fixed element of S,. Then the map w :A(UJ-,A(S,) defined by

v( c f (t)x’)(x.) = {Z ISTO

Et,UeZise


is an algebra map. Moreover, since for each I E TO {t} $ IF, W(x’)(_q) = 0. Thus

Conversely suppose that $J : A(UO) + A(S 0 is not a dual map. Thus, there exists ) s E SO such that FS is not principal. Let (Xh)hcH, H E K, be a partition of TO and let f : TO-, K be the map defined by f(t) = h whenever t E X,,. We have: if ~(&,f(t)x’)(x~) = k, then X, E IF, and X,, 4 ES for each h # k. In fact

w( x V(r) - W)(x,) = v( x fW+J - kv( c +s, = 0. ST” IET” lETO

Therefore, by Proposition 3.23, X, E IFS and, by Lemma 3.24, US is IKI+- complete. 0

We conclude our observations about the relationships between algebra and coalgebra maps with a proposition which specifies the links that must intervene between lG] and IK:I so that every algebra map ‘1’ :A(UJ+A(S”) is dual of a coalgebra map. If ITo] is a measurable cardinal (i.e. there exists a non principal (T,(-complete ultrafilter in P(T,)), then every algebra map rj.~ :A(%,)-, A(&,) is continuous if and only if ITO1 6 IKl; while, if ITO] is not a measurable cardinal then it is sufficient that ITO] =S JKI+. In any case, if there exists a cardinal LY such that a < ITO] < 2” then the condition IKI 2 cy is sufficient so that every algebra map r+~ :A(UO)+A(S,,) is dual of a coalgebra map, as we deduce by:

Proposition 3.26. Zf ITo1 s 21K’ then every algebra map + :A(UJ-,A(&,) & dual of a coalgebra map.

Proof. It is a trivial consequence of Proposition 3.25 and of the following classical result. 0

Proposition 3.27. Let A, B be sets. Zf (BI c 21A’ then there exists no IA I+-complete non principal ultrafilter in P(B).

Combining the previous propositions we obtain a generalization of Leroux’s result (see [S]) about the continuous endomorphisms of the Incidence Algebra of a finitely generated Mobius Category.

Proposition 3.28. Let U be a finitely generated decomposition structure. Then every automorphism of A(U) is continuous. Zf ( T,,I c 21K’ then every endomorphism of A(U) is continuous.

The first result of this kind was obtained by Baclawski in [l] for Incidence Algebras of locally finite posets. In the same paper it is shown that: “Every derivation of an Incidence Algebra of a locally finite poset is continuous”.

144 L. Azzena, F. Pirm

Now, we want to prove a similar result for Incidence Algebras of finitely

generated decomposition structures.

We shall denote D(S) the space of the K-linear maps D :A(S)+A(S) satisfying, for all f, g E A(S),

D(f*g)=f*D(g)+D(f)*g.

The elements of D(s) will be called K-derivations (or simply derivations) of

AC+).

Proposition 3.29. Let D be a derivation of A(S). Then, for each idempotent element f, D(f) E J,(S).

Proof. Let f be an idempotent element of A(S). If I E SO then

D(f )(x,) = D(f *f )(xr) = 2f (xr)D(f )(xr).

Therefore, since f(q) # 4, D(f )(x,) = 0. 0

A derivation D of A(S) will be said to be an inner derivation with respect to

g EA(S) if, for each f E A(S), D(f) =g *f -f *g. The inner derivation with

respect to g will be denoted Dg. The following proposition about the inner derivations is central. Its proof is

inspired by that of a proposition due to Baclawski [l].

Proposition 3.30. Let S be a hereditarily finite decomposition structure, and D a derivation of A(S). Then there exists an inner derivation Dg such that D(Y) = Dg(xU) for each u E S,.

Proof. Notice that (D(x”) *xS),,% is a summable family, thus g = Csc$ D(Y) *xs

is an element of A(S). Now, if u E S, we have:

Dg(xU)=g*xU-xU*g=D(xU)*xU- c xU*D(xS)*xS. SQSo

Since

xU*D(xS)*xS = -D(xU)*xS if u fs o

if u =s,

DR(xU) = D(x”) *xU+ 2 D(x”)*x”=D(x”)*x”+D(x”)*(~-2’) S#U

= D(x”). 0

Coronary 3.31. Let S be a hereditarily finite decomposition structure and D a

derivation of A(S). Zf D(x”)(x~) Z 0 then

or a,(s)= i%(r) and h(s) E [4(r), W)l

a,(s) = h(r) and a,(s) E [4drh 4(r)l-

Incidence algebras and coalgebrm 145

Proof. If we put Do = D - Dg and if D(x”)(x~) # 0 we see that:

or

Q,(~“)(~,) = (x M’) * D”(Y) * x~~‘~‘)(x~) # 0.

Thus the corollary holds. 0

Let us now suppose that s is a finitely generated decomposition structure.

Under this hypothesis we prove a proposition similar to Proposition 3.14.

Proposition 3.32. Let s be a jinitely generated decomposition structure. If D is a derivation of A(S) then, for each pair u, u E S, and for each n 2 1

D(x” *J,@) *xv) G (51(s))“-‘.

Proof. If f E xU * J&) *x “, by Proposition 3.6, f is a finite sum of expressions

such asx”*f,**-.*f,*x”, h EJ~(!S). Thus, D(f) is a finite sum of expressions

like

D(x”) * fi * - . .*fn*xU+xU*D(fi)*. . .*fn*xU+. . .

+x”*fi** ..*fn*D(xU)

where every addendum is in (.Ji(s))“-‘. q

Corollary 3.33. Under the hypotheses of Proposition 3.32 if D(Y)&.) # 0 then I(s) < l(r) + 1.

Proof. Let r be an element of S such that D(Y)(q) # 0 and l(r) = m. If l(s) = n then xs EX~‘)@)*J~(~) *x ‘l(‘). Therefore D(x”) E (J1($5))“-‘. Thus, if m < l(s) - 1

then D(x”)(x~) = 0. 0

Finally we can prove our main result about the derivations of A(S).

Proposition 3.34. Let s be a finitely generated decomposition structure. If D :A(s)+A(s) is a derivation then D is continuous.

Proof. By Proposition 3.30, we can write D = Dg + Do where g = Es+ D(x”) *xs. Since Dg, like every inner derivation, is continuous, to conclude the proof it is

sufficient to prove that Do is a continuous derivation. At first we observe that if

r E S, and D,(x”)(x,) # 0 then (xJo@) * D,(x”) *x’~(~))(x,) # 0. Therefore by

Corollary 3.33, the elements s E S such that D,(x”)(x~) # 0 are elements of

S,+,(E+,(r), a,(r)) which is a finite set. Now, let (k,),,s be a family of elements of


K. We have:

Since

= Do xJo(r) * 2 k$*x SE.7

J”“)w = “(..s(Jg,J,(.,, k,x”)W = Do ( c

~Gn+l(JO(~).J1(~)) ksx”)(xJ + Do(..cs-s*+~“(r),J,(r)) k,x’)(xJ- c k,x” l x~~(‘)*.f,,,+~(S) *x+@),

~~(~-~m+l)(w~)~ J,(r))

by Proposition 3.32,

& c kx” > E Jm+d%. sc(s-s,+l)(ao(r),al(r))

Hence

Do( x W)(xJ = Do( c kx” (xr) sss ~an+l(Jo(~).Jl(~))

) = c WoW)(x,) = c WoW)(xr). 0

~4n+l(Jo(~)~Jl(~)) SES

References

[l] K. Baclawski, Automorphisms and derivations of Incidence algebras, Proc. Math. Sot. 36 (1972)

351-356. [2] C.C. Chang and H.T. Keisler, Model Theory (North-Holland, Amsterdam, 1973).

[3] M. Content, F. Lemay and P. Leroux, Categories de Mobius et fonctorialites: un cadre general

pour l’inversion de Mobius, J. Comb. Theory, A 28 (1980) 169-190.

[4] P. Doubilet, G.-C. Rota and R.P. Stanley, On the Foundation of Combinatorial Theory VI: The

Idea of Generating Function. In Finite Operator Calculus (Academic Press, New York, 1975)

83-134. [5] A. Dur, Mobius functions, incidence algebras and power series representations, Lecture Notes in

Math. 1202 (Springer Berlag, New York 1986).

[6] S.A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math 61

(1979) 93-139. [7] A. Joyal, Une thCorie combinatoire des series formelles. Ad. Math. 42 (1981) l-82.

[8] P. Leroux, The isomorphism problem for incidence algebras of Mobius categories, Illinois J. Math. 26(1982) 52-61.

[9] R.P. Stanley, Structure of incidence algebras and their automorphism groups, Bull. Amer. Math.

76 (1970) 1936-1939. [lo] M.E. Sweedler, Hopf Algebras (Benjamin Inc. New York 1969).

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